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Questions tagged [cardinals]

This tag is for questions about cardinals and related topics such as cardinal arithmetics, regular cardinals and cofinality. Do not confuse with [large-cardinals] which is a technical concept about strong axioms of infinity.

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1answer
13 views

Subdirectly irreducible algebra of language F with cardinality $>=2^\kappa$

If language F has cardinality $\kappa$ ($\kappa$ is some infinite cardinal) and arity of every operation in F is 1, then doesn't exist subdirectly irreducible algebra of language F with cardinality $&...
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1answer
17 views

Cardinality: Injection between subsets of Uncountable set

assuming, S is infinite uncountable, I am trying to come up with injective f: (S union N) -> S. Where N is naturals. So far I created S0 which consists of infinite sequence of elements of S, such that ...
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0answers
40 views

Cardinality of an uncountable set after union with another uncountable set, that has smaller cardinality.

When $A$ is uncountable and $B$ is a countable set, $|A\setminus B|=|A|$. How can I prove (or disprove) that $|A\setminus a|=|A|$ or $|A\cup a|=|A|$, where $A, a$ are uncountable sets and $|A|>|a|$ ...
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2answers
41 views

Set $A$ is countably infinite if and only if there exists a bijection $f: \mathbb{N} \rightarrow A$

Using the fact that for any $A$, $A$ is countably infinite if there exists a bijection $f: A \rightarrow \mathbb{N}$, how do I prove the statement: $A$ is countably infinite if and only if $\exists$ ...
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2answers
58 views

Prove that $|\mathbb{N}\times \mathbb{R}| = |\mathbb{R}|$

I'm trying to prove the equality between the cardinalities: $|\mathbb{N}\times \mathbb{R}| = |\mathbb{R}|$ ($\mathbb{N}\times \mathbb{R}$ is cartesian product from $\mathbb{N}$ to $\mathbb{R}$). I ...
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0answers
32 views

linearly ordered family of sets cardinality greater than supremum of individual sets

Does there exist a set $X$ linearly ordered by $\subset$ where $|\bigcup X|>\sup\{|x|:x\in X\}$? I'm having trouble thinking around constructing one. I'm fairly certain that finite sets won't work,...
1
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1answer
60 views

Is there any bijection between $\mathbb{R}$ and $\mathbb{R}^2$? [duplicate]

Is there any bijection between $\mathbb{R}$ and $\mathbb{R}^2$ ? If have then what is the mapping ? Please define the mapping. They have same cardinality then it is possible to have a bijection ...
2
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1answer
80 views

Using union of countably infinite sets, I tried to prove that set of all real numbers in [0,1) is countable

Cantor's diagonal method shows that the set $S=\{x\in \Bbb R|x \in [0,1)\}$ is uncountably infinite, because there is no bijection between the set $S$ and the set of natural number $\Bbb N$. I came ...
1
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1answer
36 views

Show $\left| \{H,T\ \}^{\oplus \mathbb{N} }\right| = \left| \mathbb{R}^{\mathbb{N} }\right|$

Let $\mathbb{R}^{\mathbb{N}} = \{ f: \mathbb{N} \to \mathbb{R}\}$ with the product $\sigma$-algebra $\mathcal{B}^{\otimes \mathbb{N}} = \bigotimes\limits_{n \in \mathbb{N}} \mathcal{B}$, where $\...
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1answer
35 views

$\kappa\cdot\sum_{i\in I}\lambda_i=_c\sum_{i\in I} \kappa\cdot \lambda_i$

Moschovakis Exercise x4.20 Prove that for all indexed families of cardinals, $$\kappa\cdot\sum_{i\in I}\lambda_i=_c\sum_{i\in I} \kappa\cdot \lambda_i$$ We have $$\kappa\cdot\sum_{i\in I}\lambda_i=...
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0answers
28 views

$\lambda\le_c\mu\implies \kappa^\lambda\le_c\kappa^\mu$

Moschovakis, (part of) Exercise x4.16: Prove that for all cardinal numbers $\kappa,\lambda,\mu$ $$\lambda\le_c\mu\implies \kappa^\lambda\le_c\kappa^\mu$$ provided $\kappa\ne 0$. $\le_c$ means "...
1
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1answer
31 views

order type of club - cofinality

With $ot(C)$ being the order-type of the club $C$, and $cof(\alpha)$ the cofinality of the ordinal $\alpha$ Show that for every limit ordinal $\alpha$ there is a club $C\subseteq \alpha$ such that $...
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2answers
52 views

How can I prove that if $\alpha$ is an ordinal, then there is an initial ordinal $\kappa$ such that $|\alpha|=|\kappa|$?

I'm having trouble understanding initial ordinals. In particular, I can't prove a seemingly trivial theorem about them. Def: An ordinal $\kappa$ is an initial ordinal iff $\forall \delta < \kappa \...
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0answers
57 views

Proving $|A\cup B|\le_c |A|+|B|$

I'm using the notation from this question. For all sets $A,B$, $|A\cup B|\le_c |A|+|B|$, and if $A\cap B=\emptyset$ then $|A\cup B|=_c |A|+|B|$. My attempt to prove it: By definition, $$|A|+|B|=||...
1
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1answer
59 views

Problem of cardinal assignment

A weak cardinal assignment is any definite operation on sets $A\mapsto |A|$ which satisfies (C1) and (C3), and it is a strong cardinal assignment if it also satisfies (C2). The cardinal numbers (...
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0answers
17 views

Set $X$ such that $2^c \leq |X|\leq |(\ell^\infty)^*|$ [duplicate]

I want to prove that $2^c\leq|(\ell^\infty)^*|$. My question is: Is there a set $X$ with $2^c\leq|X|$ such that $f:X\to(\ell^\infty)^*$ is an injective function or $g:(\ell^\infty)^*\to X$ is a ...
0
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1answer
19 views

S is a possibly infinite set. Prove |S| < |P(S)|

Suppose S is a set. Do not assume that S is finite. How can one prove that |S|<|P(S)|? (P(S) is the power set of S). Would I say something how the power set is the subset of S so that it contains ...
3
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1answer
31 views

Diagonal argument applied to computable numbers

Upon applying the Cantor diagonal argument to the enumerated list of all computable numbers, we produce a number not in it, but seems to be computable too, and that seems paradoxical. For clarity, ...
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0answers
38 views

for any infinite set A, |A| >= |N| [duplicate]

I try to prove that the power of any infinite set must be equal/bigger then the power of the natural א0. I tried to show that for any infinite set there exists a subset of it, that its power is the ...
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0answers
108 views

Two well orderings of an infinite cardinal agree on a large set

I've seen this question but I'm having trouble following the proof given. This is an exercise from Kunen: If $\kappa$ is an infinite cardinal and $\triangleleft$ is a well ordering on $\kappa$, then ...
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0answers
26 views

$\sigma$-ideal of subsets of $2^\omega$

I do not understand why here on the page 148 $\cal M^*_{2, K}$ is a $\sigma-$ideal of subsets of $2^\omega$. I even do not know the weaker statement: why it is an ideal?
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1answer
13 views

$X \subset \mathbb{R}_{>0}$ exists $C$ for every $\{x_1, …, x_n\} \subset X$ $\sum^{n}_{i=1} x_i<C$ Then $X$ is countable

Let $X \subset \mathbb{R}_{>0}$ such that exists $C$ that, for every $\{x_1, ..., x_n\} \subset X$ $$\sum^{n}_{i=1} x_i<C.$$ Prove that $X$ is countable My intuition goes that if $X$ is not ...
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4answers
65 views

Cardinality of $A$ and the Natural Numbers [closed]

The question asks to show that the set $A$ and the set of natural numbers have the same cardinality. Where $$A=(1,2,3)\times\mathbb{N}$$ So I know I have to prove a bijection, but I am having ...
2
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2answers
62 views

cardinality proof: prove that for any set $A, \ 2^{|A|} \neq |\mathbb{N}|$ [duplicate]

Prove: for any set $A, \ 2^{|A|} \neq \aleph _{0}$ as $\aleph_{0} = |\mathbb{N}|$. my attempt: Suppose by contradiction that there exist a set $A$ such that $2^{|A|} = \aleph_{0}$, which Implies ...
0
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1answer
49 views

Decomposition of an $\aleph_1$ set into $\aleph_1$ sets $\aleph_1$ [closed]

Can we justify the claim Any set of cardinality $\aleph_1$ can be expressed as the disjoint union of $\aleph_1$ sets of cardinality $\aleph_1$ in a simple yet reasonably-correct way?
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1answer
29 views

An ordinal equipotent to a natural number is equal to that number

Let $\alpha, \beta$ be ordinals and $n$ be a natural number. How can one prove the following $$ \forall \alpha (\alpha \approx n \implies \alpha = n)$$ using the fact $$ \vert \alpha \vert \le \beta \...
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1answer
36 views

what is the cardinality (or algebraic dimension) of a hilbert space over a cardinal $\kappa$?

Something has been bugging me lately; maybe you guys can help satisfy my curiosity. Let $\kappa$ denote any nonzero cardinal. For convenience, we identify $\kappa$ with its smallest ordinal ...
3
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1answer
183 views

Why is the definition of cardinal number as the set of all sets equivalent to a given set “problematical”?

In Fundamentals of Mathematics, Volume 1 Foundations of Mathematics: The Real Number System and Algebra, after defining set equivalence as the ability to put the elements of the related sets in one-to-...
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0answers
37 views

Cardinal number of the Banach space $c_0(A)$

For infinite dimensional Banach space $X$, the cardinality of $X$ is equal to its algebraic dimension. Now let $A$ be an infinite set and define $c_0(A)=\{f:A\rightarrow\Bbb R, f$ is bounded and $ \...
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1answer
37 views

cardinality of all cardinal numbers less than a cardinal number

For a cardinal number $\alpha$ what is the cardinality of the set $X=\{\beta, \beta$ is a cardinal number with $\beta<\alpha\}?$ Can we say for example that $card(X)=\alpha$ or $card(X)\leq \...
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0answers
43 views

Cardinality of all cardinal numbers less than a given cardinal

For a given cardinal number $\aleph_{\alpha}$ we define $$X_{\alpha}= \{\aleph_{\beta}; \aleph_{\beta}<\aleph_{\alpha}\}.$$ We can easily prove that 1) $card(X_{\alpha}) \leq \aleph_{\alpha}^{+}=...
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1answer
72 views

Value of the cardinal product $\aleph_1 \cdot \mathfrak{c}$

Suppose we want to know how many well-orderings of the naturals there are. That is, not up to isomorphism, but how many individual ways to well order the naturals there are. It's easy to see that for ...
1
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1answer
28 views

Set of various order types of a set

Starting from the cardinal $|\Bbb N| = \aleph_0 = \beth_0$, we can generate a larger cardinal in two ways: Take the set of all subsets, generating the cardinal $\beth_1$ Take the set of all well-...
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0answers
29 views

The “number” of complete vector fields in a manifold

I was wondering about the cardinality of complete vector fields in relation to the cardinality of general vector fields. Let $$\mathcal{A}=\{X:M\rightarrow TM| X \text{ is a vector field}\}$$ and $$\...
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1answer
45 views

Question about the existence of sets

If there is a sequence of set $a_{i}$ and $i \in I \wedge |I|>|\mathbb{R}|$, is the set $A$ which $\forall i (i \in I \rightarrow a_{i}\in A)$ and the set $\bigcup \limits_{i \in \mathbb{R}} a_{i}$ ...
1
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1answer
49 views

Are infinite indexes multiplicative?

For a group $G$ and its subgroup $H$, the index of $H$ relative to $G$, denoted by $[G:H]$, is the cardinality of the set $\{ gH \mid g \in G \}$. It is known that $|G| = [G:H]|H|$ even if all ...
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1answer
29 views

Explicit example of $C \subset \mathbb{R}$ with a single certain type of condensation point… if such is even doable.

In Abbott's Understanding Analysis 2e, part of exercise 1.5.10 asks Let $C \subset [0,1]$ be uncountable, let $A := \{ a \in (0,1): C \cap [a,1] \text{ is uncountable} \}$, and let $\alpha := \sup ...
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1answer
78 views

Find the cardinality of $A= \left\{ f : \mathbb N \rightarrow \mathbb N \mid \forall x:f(x)\le x\right\}$

How can I solve tasks like this one? Example task Find the cardinality of $$A= \left\{ f : \mathbb N \rightarrow \mathbb N \mid \forall x:f(x)\le x\right\}.$$ I know that $|A| \le \mathfrak{c}$...
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2answers
66 views

Show that the cardinal of $ A := \left\{ k \in \mathbf{Z} | 0 \leq k \leq n \text{ and } \binom{n}{k} \text{ is odd} \right\} $ is a power of 2 [duplicate]

Let $ A := \left\{ k \in \mathbf{Z} | 0 \leq k \leq n \text{ and } \binom{n}{k} \text{ is odd} \right\} $. I must show that the cardinal of $A$ is a power of 2. I have tried to show that there ...
2
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1answer
41 views

How to prove $\mathbb{R}^n < \mathbb{N}^\mathbb{R}$

I know that the cardinality of $\mathbb{R}^n$ is equal to $\mathbb{R}$. I also know that the cardinality of $\mathbb{N}^n$ is equal to $\mathbb{N}$, but how do I prove that the cardinality of $\...
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2answers
56 views

Does it make sense to say that “there are more non-Abelian groups than Abelian groups”?

Firstly, does the family of all non-isomorphic non-Abelian groups have a well defined cardinality? How about the family of all non-isomorphic Abelian groups? If they are both defined, how do they ...
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0answers
24 views

cardinality of sets intersections

Given that: $$| A \cap \mathbb{R}| = |\mathbb{R}| $$ $$| B \cap \mathbb{R}| = |\mathbb{N}| $$ Can we say that: $$|A| = |\mathbb{R}|$$ $$|B| = |\mathbb{N}|$$ ? If not, can we say something for ...
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0answers
15 views

Cardinality of set of words [duplicate]

How can we prove that for the countable alphabet $A$ the cardinality of set of all finite words over this alphabet $A^*$ is equal to $\aleph_0$
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0answers
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Is the least inaccessible cardinal equivalent to the first aleph fixed point? [duplicate]

Let $I$ be the least / first inaccessible cardinal. As inaccessible cardinas are all aleph fixed points, and they are regular, so each inaccessible cardinal is an aleph fixed point after the previous ...
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2answers
53 views

Cardinality of sets of functions $\mathbb{R}\to \mathbb{N}$ and $\mathbb{N}\to \mathbb{R}$. [closed]

Let $B^A$ denote the set of all functions $A \to B$. Prove that $\left|\mathbb{R}^\mathbb{N}\right|<\left|\mathbb{N}^\mathbb{R}\right|$.
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1answer
18 views

$\operatorname{card}(\bigcup\limits_{n \in \mathbb{N}} \underbrace{A\times…\times A}_{n})=k$ if $\operatorname{card}(A)=k$ infinite.

I was reading a proof of a theorem that goes like this: Let $A$ be an infinite set of cardinality $k$ and $A^{<\omega}$ the set of finite sequences of elements of A. Then $\operatorname{card}(A^{&...
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1answer
20 views

Proving sets of functions have same cardinality

Prove that $$card(A^{B \times C})=card(A^{B^C})$$ where $A^B$ is a set of all functions from $B$ to $A$ and $A \times B$ is cartesian product of sets. Is the bijection supposed to be sending $f$ to $...
0
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0answers
71 views

Cardinality of infinite dimensional vector space

Assume that V is an infinite dimensional vector space. I know that if V is a vector space over a field F, then |V|=max{dimV,|F|}. So if we take V=$\mathbb{R}$ and F=$\mathbb{Q}$ then |V|>|F| and |V|=...
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0answers
16 views

Cardinality of $[\lambda]^\kappa$

Let $\kappa \leq \lambda$ cardinals with $\lambda$ infinite, and $[\lambda]^\kappa=\{Y\subseteq\lambda : ot(Y,\in)=\kappa\}$. I want to show that $[\lambda]^\kappa \asymp\ ^\kappa\lambda$. I've ...
0
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2answers
36 views

How to show $\operatorname{card}(\omega+1)=\omega$

Apparently $\operatorname{card}(\omega+1)=\omega$. This means that there is an order $<$ on $\omega+1$ such that there is an isomorphism of ordered set $f$, $(\omega+1,<) \cong (\omega,\in)$, ...